Free surface flow focusing

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Free surface flow focusing

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Prof. dr. G. van der Steenhoven (voorzitter) Universiteit Twente

Prof. dr. D. Lohse (promotor) Universiteit Twente

Dr. D. van der Meer (assistent-promotor) Universiteit Twente

Prof. dr. A.P. Mosk Universiteit Twente

Prof. dr. C. Clanet LadHyX, France

Dr. R. Hagmeijer Universiteit Twente

Prof. dr. J.J.W. van der Vegt Universiteit Twente

Prof. dr. J.M. Gordillo Universidad de Sevilla, Spain

The work in this thesis was carried out at the Physics of Fluids group of the Faculty of Science and Technology of the University of Twente. Financial support for this work was provided by the Dutch Organization for Scientific Research (NWO). Nederlandse titel:

Convergerende stromingen en vrije oppervlakken

Publisher:

Ivo R. Peters, Physics of Fluids, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands pof.tnw.utwente.nl

ivo.r.peters@gmail.com Cover design:

Ineke Koene

c

Ivo R. Peters, Enschede, The Netherlands 2012 No part of this work may be reproduced by print, photocopy, or any other means without the permission in writing from the publisher

ISBN: 978-90-365-3376-8 DOI: 10.3990/1.9789036533768

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FREE SURFACE FLOW FOCUSING

PROEFSCHRIFT ter verkrijging van

de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,

Prof. dr. H. Brinksma,

volgens besluit van het College voor Promoties in het openbaar te verdedigen

op vrijdag 29 juni 2012 om 16.45 uur door

Ivo Remco Peters geboren op 10 februari 1984

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Prof. dr. rer. nat. Detlef Lohse en de assistent-promotor: Dr. Devaraj van der Meer

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1

Introduction

Many physical and chemical processes require a high energy density to run, or to keep running. An efficient way to achieve high energy densities is to start with a low energy density and to focus this into a small volume. According to legend the Greek scientist Archimedes used this principle during the siege of Syracuse, by focusing sunlight onto Roman ships in an attempt to set them to fire. In modern science, we are still trying to push the limits to which we can focus energy. A prime example is the development of nuclear fusion reactors, where the key ingredient is to confine a plasma at such a high energy density that nuclear fusion reactions take place. A possible way to achieve this, is to generate a focusing flow of plasma [1, 2]. The most extreme examples of flow focusing are actually happening naturally and are found many lightyears away, for example in the gravitational collapse of stars. These two examples are just a few of many others that require thorough understanding of focusing flows, which is the main subject of this thesis.

Incompressible flows that are dominated by inertia have the property that focus-ing of the flow results in an increase of the velocity. This property is very commonly applied, for example in the summer by children narrowing the end of a garden-hose to convert the rather weak stream that flows out usually, into a strong jet that reaches much further. To describe these flows, one has to recognize that the flow rate (volume per unit of time) in these flows remains constant. This constant flow rate condition then tells us that the smaller the area A trough which a fluid has to flow, the larger the velocity U becomes:

U ∝ A−1. (1.1)

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(a) (b)

Figure 1.1: (a) a converging-diverging “de Laval” nozzle, where gas is accelerated to supersonic speeds. (b) after the impact of a disc on a water surface, a surface cavity is formed which acts as a liquid nozzle that accelerates the gas flow inside the cavity. The arrows in both images indicate the direction of the gas flow.

This principle is not only used by people trying to clean their car or motorcycle with a garden hose, but also in rocket engines where exhaust gasses are accelerated to speeds well above the speed of sound by a converging-diverging nozzle [3] as shown in Fig. 1.1(a). In the latter example however, one has to be aware that the speeds become too high for Eq. (1.1) to hold due to compressibility∗.

In the examples mentioned above, a fluid flow is focused by means of a solid noz-zle, that decreases in surface area downstream. In the case of a gas stream however, it is also possible to create a similar focusing nozzle from a liquid. Fig. 1.1(b) shows that such a nozzle is formed naturally after the impact of a solid on a liquid surface, where hydrostatic pressure is driving the gas flow, and the shape of the free surface is converging the flow through a small area. One becomes aware of the complexity of this system, when realizing that the liquid is not only influencing the gas stream, but the gas also starts to have its effect on the liquid: The result is a system with a subtle interplay between the dynamics of the gas and the liquid phase.

The free surface of a liquid can act as a boundary to focus a second fluid, but can also become the subject of flow focusing itself. A particular example is the pinch-off of an air bubble in a liquid, for example from an underwater nozzle [4–7] or after the impact of a liquid or a solid on a liquid surface [8–10]. In these cases, a void is

∗_{Compressibility needs to be taken into account for Ma > 0.3, where Ma is the Mach number. After}

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3 created inside a liquid, which shrinks due to capillary forces or hydrostatic pressure. Eventually, the void pinches off, where velocities diverge as the moment of pinch-off is approached, and the collapse becomes inertia-dominated. This divergence of velocities is the result of focusing the liquid flow towards a single point: the area A through which the liquid can escape approaches zero, which, according to Eq. (1.1) leads to U → ∞.

A collapse driven by inertia is of interest for many applications because it can be a very violent phenomenon: The jets that are formed after the collapse of cavitation bubbles are responsible for the damage of ship propellers [11, 12], but can also be used to pulverize kidney stones [13, 14] or to remove dentine debris from root canals [15, 16]. The approach to a singularity is of great fundamental interest, especially because after recent experiments it was thought that, unlike the pinch-off of a liquid drop which is very well described by a universal law [17, 18], the pinch-off of an axisymmetric air bubble was non-universal [10, 19, 20]. These systems are different because the liquid drop pinch-off is governed by surface tension, where the bubble pinch-off is inertia-driven. The discussion was settled thereafter by showing that the axisymmetric pinch-off of an air bubble asymptotically approaches universality, but that this universal behavior often is hidden: something like self-similarity is there much earlier, but converges slowly to the real self-similar regime [21–23]. Thus, in the case of an axisymmetric pinch-off, the collapse can be described by scaling laws independent of the specific system, which means that the system has no memory of its initial state. A slight deviation from axisymmetry however, again changes the whole picture: it was found that even the smallest deviation from axisymmetry is remembered until the pinch-off [24–26]. A peculiar aspect of this memory of the initial state is that the absolute amplitude of the perturbation is remembered: the relative amplitude therefore grows towards the pinch-off, which results in the fact that eventually the perturbation dominates the shape of the free surface, and also determines the way the bubble pinches off.

In industrial applications flow focusing can be both your enemy or your friend, which we illustrate below with a few examples of both cases.

Situations where a free surface is driven to very small areas can sometimes be limiting in applications, like e.g. immersion lithography [27–29]. In immersion lithography a droplet is placed between a lens and a wafer to increase optical res-olution, and this droplet stays attached to the lens as it moves over the wafer surface. Above a certain speed a sharp corner is formed at the tail of the droplet, which even-tually leads to detachment of small droplets if the speed is increased further [30, 31]. To prevent the undesired deposition of small droplets on a wafer, one would like to avoid the formation of strongly curved interfaces. The first step for this would be to understand exactly why such a sharp corner is formed, and how the size and shape of the corner can be predicted.

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50 μm

Figure 1.2: A jet with a typical diameter of 5 µm, traveling at a speed of 490 m/s. Time between the images is 500 ns. Courtesy of A. van der Bos, N. Oudalov, Y. Tagawa and C.W. Visser.

Strongly curved surfaces can however also be very useful in applications. For needle-free injectors, it is necessary to create a fast, thin jet [32]. Making use of the strongly curved meniscus that is naturally formed in a capillary, an ultrafast jet can be generated by shooting a laser in the liquid [33], see Fig. 1.2. In this case, the curved free surface acts as a focusing mechanism for the flow that was created by laser-induced cavitation [34, 35]. The formation of jets from a meniscus is also widely applied in ink-jet printing [36, 37], where the flow is driven by a piezo-electric element.

The objective of this thesis is to obtain a better understanding of the subjects described above. We will do so by investigating the topics described in the following chapters:

In Chapter 2 and 3 we investigate the role of air inside a collapsing cavity created by the impact of a round disc on a water surface. For this we apply three different methods to measure the air flow. In Chapter 2 we first experimentally determine the contour of the cavity during formation and collapse, from which we derive the cavity volume. Secondly, we introduce smoke particles inside the cavity which we illu-minate using a laser sheet. Using particle image velocimetry (PIV) techniques, we determine the velocity of the air in the cavity. Both direct and indirect measurements agree very well and comparing our measurements with boundary integral (BI) simu-lations gives excellent agreement. We find that, just before pinch-off, compressibility of the air plays an important role in the dynamics of the cavity. In Chapter 3 we show that the air inside the cavity can even reach supersonic speeds.

In Chapter 4 we replace the circular disc that was used in Chapters 2 and 3 by one that has a non-axisymmetric shape similar to the petals of a flower. For small harmonic disturbances we closely follow how these disturbances grow and oscillate

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5 during the collapse of the cavity. Our experimental results compare excellently to theoretical predictions, and ultimately, by solving the collapse of the cavity in (un-coupled) horizontal layers, were able to completely reconstruct the three-dimensional shape of a cavity that was created by the impact of a disc with a mode-20 harmonic disturbance. Increasing the amplitude of the perturbations, we depart from the linear theory, and find astonishing non-linear effects like the formation of sub-cavities and secondary jets.

In Chapter 5 we study the shape of a splash that is created in the very first in-stances of the impact of a circular disc. Using experimental observations and bound-ary integral simulations we show that the splash exhibits a self-similar behavior for any value of the Weber number, the dimensionless quantity that compares inertia with surface tension. We show that there exists a critical Weber number, above which small droplets are ejected from the rim of the splash, we show that a Rayleigh-Taylor instability is responsible for this transition.

In Chapter 6 we introduce a second liquid phase in our system by creating an experimental setup with three flowing components: Water, oil and air. By impacting the disc on a layer of oil that is floating on a deep layer of water, we obtain a deep understanding of the jet formation and the bulk flow after cavity collapse, which also applies for the single-liquid case. Specifically, we experimentally prove that the jet is created from the surface of the cavity, confirming earlier theoretical predictions [38]. By using a deep layer of oil and starting the disc at the oil-water interface, we create a two-fluid system without air, where the role of gravity has been greatly reduced. Pulling the disc down entrains a column of oil into the water; we show that the shape of the entrained oil becomes universal for high disc velocities.

In Chapter 7 we numerically investigate the formation of micro-jets created by laser induced cavitation. We perform boundary-integral simulations that closely re-produce experimentally obtained results. Using the insight obtained with numerical simulations, we develop a simple analytical model that accurately predicts the jet velocity dependence on the relevant parameters.

In Chapter 8 we investigate millimeter-sized silicone-oil drops that are sliding down an inclined surface. Drops that are sliding faster, tend to obtain a cornered shape at their tail. We experimentally show that the curvature at the corner increases exponentially with sliding speed. We explain this exponential increase by showing that the nanometric cut-off length, related to the classical viscous singularity at a moving contact line, plays an essential role in the selection of the curvature at the tail of a sliding drop.

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[22] S. Gekle, J. H. Snoeijer, D. Lohse, and D. van der Meer, Approach to universal-ity in axisymmetric bubble pinch-off, Phys. Rev. E 80, 036305 (2009).

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2

Air flow in a collapsing cavity

∗

We experimentally study the airflow in a collapsing cavity created by the impact of a circular disk on a water surface. We measure the air velocity in the collapsing neck in two ways: Directly, by means of employing particle image velocimetry of smoke injected into the cavity and indirectly, by determining the time rate of change of the volume of the cavity at pinch-off and deducing the air flow in the neck under the assumption that the air is incompressible. We compare our experiments to boundary integral simulations and show that close to the moment of pinch-off, compressibility of the air starts to play a crucial role in the behavior of the cavity. Finally, we measure how the air flow rate at pinch-off depends on the Froude number and explain the observed dependence using a theoretical model of the cavity collapse.

2.1 Introduction

The impact of a solid body on a water surface triggers a series of spectacular events: After a splash, if the impact speed is high enough, a surface cavity is formed which pinches off such that a bubble is entrained [1–3]. Right after pinch-off two strong thin jets are formed [4], one shooting upwards and one shooting downwards.

An aspect in the impact on liquids that has drawn particularly very little attention is the influence of the accompanying gas phase. When we take into account the inner gas in the detaching air bubble, we find a singularity in the velocity of the inner

∗_{Ivo R. Peters, Stephan Gekle, Detlef Lohse, and Devaraj van der Meer, Air flow in a collapsing}

cavity, Preprint (2012)

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gas. Assuming any finite flow rate for the gas, the velocity of the gas will diverge because the area that the gas has to flow through goes to zero. Nature has found a way to avoid a true singularity by letting compressibility limit the speed of the air, but nonetheless the air plays an important role in the final shape of the cavity just before pinch-off [5, 6], and can even reach supersonic speeds (see Chapter 3).

The main objective of this chapter is to understand what determines the gas flow rate in the case of an impacting disc and to obtain insight in the role of compressibil-ity effects in the air. To this end we apply two different approaches: First we perform volume measurements to determine the flow rate based on continuity, and second we measure the air flow directly by seeding the air with smoke and laser sheet illumina-tion. We compare and extend our experiments with numerical simulations, where we use one- and two-phase boundary integral simulations, sometimes coupled to com-pressible Euler equations [7], to determine the air flow, with and without taking the dynamics of the gas phase into account.

We have structured this chapter as follows: We first give a brief description of the experimental setup in Section 2.2. Section 2.3 explains the method of volume mea-surements, and the results are combined with numerical simulations. More specifi-cally, we measure how the air flow rate at pinch-off depends on the Froude number and explain the observed dependence using a theoretical model of the cavity collapse. In Section 2.4 we perform a direct determination of the air flow velocity by seeding the air with smoke and illuminating with a laser sheet. Subsequently, we compare the results with the velocities that we determined using volume measurements. Finally, in Section 2.5 we discuss in detail when and how compressibility becomes important.

2.2 Experimental setup

The experimental setup consists of a water tank with a bottom area of 50 cm by 50 cm and 100 cm in height. A linear motor that is located below the tank pulls a disc through the water surface at a constant speed. This disc is connected to the linear motor by a thin rod. The events are recorded with a Photron SA1.1 high speed camera at frame rates up to to 20 kHz. Our main control parameter is the Froude number, which is defined as the square of the impact speed U0, nondimensionalized

by the disc radius R0and the gravitational acceleration g:

Fr= U

2 0

gR0

(2.1) Two snapshots of the experiment are shown in Fig. 2.1. The left image shows the situation right after the impact. where the cavity is being formed. A downward flow of air is required to fill in the space that is created by the downward moving disk and the expanding cavity. On the right a later stage in time is shown, some moments

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2.2. EXPERIMENTAL SETUP 11

5 cm

Figure 2.1: Two snapshots of an experiment in which a disk with a radius of 2 cm hits the water surface and moves down at a constant speed of 1 m/s. A surface cavity is created that subsequently collapses under the influence of hydrostatic pressure. Eventually, the cavity pinches off at the depth indicated by the dashed line, and a large air bubble is entrained. The red arrows indicate the direction of the air flow: On the left, volume is being created, resulting in a downward air flow. On the right the bubble volume below the pinch-off depth is decreasing, and therefore air is pushed upwards.

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before the pinch-off. Here, there is a competition between the downward moving disc and the expanding part of the cavity on the one hand, and the collapsing part, i.e., the region above the maximum, on the other. The former tends to increase the cavity volume below the pinch-off depth (dashed line), whereas the latter decreases it. We always observe that close to pinch-off, the violent collapse is dominant and the bubble volume below the pinch-off point decreases, pushing air out through the neck. As the neck becomes thinner towards the moment of pinch-off, the gas speed increases rapidly. The remaining part of this chapter is devoted to measuring this air flow and comparing with numerical simulations.

2.3 Geometric approach

The first way in which we will quantify the air flow through the neck of the cavity is an indirect one: We will measure the time evolution of the volume of the cavity below the pinch-off point and calculate its first derivative with respect to time. This will be identified with the air flow rate through the neck. This involves the following assump-tions: (i) The air flow is incompressible, (ii) the air flow profile is one-dimensional (i.e., a plug flow) and only directed in the vertical direction, and (iii) the cavity shape is axisymmetric. The first assumption is only violated close to the moment of pinch-off, when the air speed diverges. Compressibility effects at this stage are investigated in Chapter 3 and its effects will be discussed in section 2.5. We will justify the second assumption partially by visualizing the air flow inside the cavity and mea-suring the velocity directly; in addition it is known from two-fluid boundary-integral simulations that the flow profile is very close to one dimensional [7]. The third as-sumption only breaks down in the neck-region very close to pinch-off because very small disturbances are remembered during the collapse (see Chapter 4 and Refs. [8– 13]). Here, this effect is only relevant locally on a very small scale and can therefore be neglected on the large scale where we measure the volume.

2.3.1 Cavity volume

We measure the volume of the cavity below the pinch-off depth as illustrated in Fig. 2.2: By tracing the contour for every frame in a movie and invoking axisym-metry we are able to determine the volume of the bubble below the pinch-off depth as a function of time. One such a measurement for a disk with radius 20 mm and impact speed of 1 m/s is shown in Fig. 2.3. In the beginning (t. 0.022 s) the vol-ume is increasing (positive slope), which means that the air at the pinch-off depth is flowing downwards. At the maximum (t ≈ 0.022 s) the flux through the pinch-off depth is zero, indicating a local stagnation of the flow at this depth. We will study this stagnation point later, in section 2.5. After this maximum the volume starts

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2.3. GEOMETRIC APPROACH 13

(a) (b) (c)

Figure 2.2: (a) The volume of the cavity below the pinch-off depth (dashed line) is determined by tracing the boundary (red line) and assuming symmetry around the central axis. (b) The volume decreases as the neck becomes thinner until the cavity closes. (c) After pinch-off a downward jet enters into the entrapped bubble, and the bubble shows volume-oscillations and cavity ripples.

0 0.01 0.02 0.03 0.04 0.05 4.5 5 5.5 6 6.5x 10 −5 t (s) V (m 3 /s) 0.04 0.045 5.5 5.6 x 10−5 t (s) V (m 3 /s)

Figure 2.3: Volume below the pinch-off depth as a function of time (blue dots), de-termined from an experiment with Fr = 5.1. The vertical dashed line indicates the moment of pinch-off. Close to pinch-off the volume decrease is well approximated by a linear fit (green line), after pinch-off the bubble oscillates with its resonance frequency (red line: fit with sine function). The steady growth in volume after the pinch-off is caused by the jet entering the bubble.

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to decrease and the flow is directed upwards. This continues until the moment of pinch-off which is indicated by the vertical dashed line in Fig. 2.3. A linear fit (green line) reveals that the flow rate is approximately constant towards the pinch-off mo-ment. More precisely, the linear fit is the time rate of change of the cavity volume at pinch-off, which is equal to minus the maximum value of volume-based flow rate, ΦV ≡ −dV /dt, under the assumption of incompressibility of the air. We will use

this maximum flow rate ΦV to compare the flow rates through the neck at different

Froude numbers.

After the pinch-off there is a clear oscillation of the volume together with a slow apparent growth of the bubble. The growth is caused by the liquid jet that is entering the bubble (Fig. 2.2c), as the amount of air is fixed after the pinch-off. Since our focus is on the behavior before pinch-off, we chose not to correct the bubble volume by subtracting this jet volume. Also, making such a correction would be complicated by the fact that the jet is imaged through the refracting, curved interface of the air bubble. Nevertheless we determined the frequency of the oscillation by fitting a sine function (red line) after correcting for the slightly positive slope. For the conditions of Fig. 2.3 the measured frequency is 143 Hz.

We compare this result with the resonance –or Minnaert– frequency f of a spher-ical bubble in water [14]: f = 3.26/r where r is the bubble radius (in meters) and the value of 3.26 m/s is based on the material properties of water. Taking the bubble volume at pinch-off, which equals 5.49 · 10−5 m3 and adopting a spherical shape to calculate the radius, we find f = 138 Hz, which is very close to the frequency of the experimentally measured volume oscillation. Note that our bubble is far from spher-ical, but it was shown that deformation of bubbles only has a small influence on its resonance frequency [15]. The agreement of the volume oscillations after pinch-off with the Minnaert frequency was also noted by [16] for the impact of freely falling objects in water.

2.3.2 Air flow rate

A characteristic quantity concerning the gas dynamics in a collapsing cavity is the air flow rate, defined as the volume of air that is being displaced per unit time close to pinch-off. From Fig. 2.3 we infer that in approach of the pinch-off point this flow rate becomes constant and can be determined as the maximum slope of the volume as a function of time (green line, Fig. 2.3), i.e., the air flow rate through the neck equals the rate of change of the volume of the cavity below the pinch-off depth, of course under the assumption that the gas flow remains in the incompressible limit. This air flow rate ΦV ≡ −dV /dt we subsequently non-dimensionalize dividing by the disk

radius squared and the impact speed (Φ_V∗ ≡ ΦV/(R2₀U0)), where the asterisk denotes

a dimensionless value. We determined the flow rate for a number of different disk radii (ranging from 15 to 30 mm) and impact speeds (0.45-1.30 m/s), the results of

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2.3. GEOMETRIC APPROACH 15 −0.50 0 0.5 1 1.5 2 0.5 1 1.5 log 10Fr log 10 Φ V * experiments simulations fit

Figure 2.4: Volume based flow rate as a function of the Froude number in a double logarithmic plot. Both the experimental data (black dots) and the numerical data (red diamonds) correspond to the maximum value of Φ∗_V. The range of experimental data is limited to Fr ≈ 12 by the appearance of a surface seal. The blue line represents the fit Φ_V∗ = 1.23Fr1/2+ 1.01.

which are shown in Fig. 2.4 where we plot the dimensionless flow rate Φ∗_V versus the Froude number Fr on a double-logarithmic scale (black dots). The experimental range is limited by the appearance of a surface seal at high impact speeds, where the crown splash is pulled inwards due to the air flow induced by the disc and closes the cavity at the surface. This surface seal usually has a significant influence on the cavity shape and dynamics [17] as well as the gas flow rate in the neck, so all of the experiments reported here are without surface seal. In the experimentally accessible regime we find an apparent power-law relation of ΦV∗ ∝ Fr0.3. When we extend the

experimental range by performing numerical simulations with our boundary integral code [7, 17], we find that the results do not lie on a straight line (Fig. 2.4, open red diamonds), which suggests that there does not exist a pure power-law.

An analytical argument- Using the assumption that the cavity expansion and col-lapse take place in horizontal non-interacting layers of fluid, an assumption that was successfully used in Bergmann et al. [17], we will now shed light onto the behavior of the air flow rate through the neck as a function of the Froude number. We will pro-vide an approximate argument in this subsection, and present a more detailed account based on the model of Bergmann et al. [17] in the appendix. For convenience from hereon we will take the z to mean the depth below the undisturbed water surface, i.e., z= 0 at the latter and increases with depth.

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Figure 2.5: The partly collapsing partly expanding cavity close to the pinch-off mo-ment can be divided in five different regions: In region I the cavity expands against hydrostatic pressure; region II is the hardly expanding/collapsing region around the maximum; in region III the hydrostatic pressure drives the collapse of the cavity, and in region IV/V continuity takes over as the driving mechanism behind the collapse. The difference between region IV and V is that the latter has a self-similar shape which is independent of the Froude number [18].

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2.3. GEOMETRIC APPROACH 17 volume ˙V= dV /dt, i.e., ΦV ≡ − dV dt = − d dt Z zdisc zc π [r(z, t)]2dz, (2.2)

where it is understood that the expression needs to be evaluated at the pinch-off time. Here, r(z,t) is the cavity profile, zdisc(t) is the vertical position of the disc and zc is

the pinch-off depth. Using Leibniz’s rule we obtain ΦV = −

Z z_disc

zc

2π r(z,t) ˙r(z,t, )dz − πR2₀U0, (2.3)

where the last term is due to the downward moving disc and ˙r ≡ ∂ r/∂ t denotes the radial velocity of the cavity wall.

To approximate the integral in Eq. (2.3) we subdivide the expanding and collaps-ing cavity –at times close to the collapse– into the regions of Fig. 2.5: In region I, just above the disc, the cavity has a radius close to the disc radius, r ≈ R0, and is

expanding against hydrostatic pressure with a horizontal velocity that is proportional to the disc velocity ˙r ∼ U0. The contribution of Region I to the integral of Eq. (2.3)

can therefore be approximated as ΦV,I ∼ −R0U0∆zI, where ∆zI is the height of

re-gion I. The second rere-gion is an approximately symmetric rere-gion, where r ≈ Rmax, the

maximum cavity radius at pinch off, and the velocity is close to zero, ˙r ≈ 0. For this reason, and also because of the symmetry above and below the vertical position of the maximum which contribute with sign change to the integral of Eq. (2.3), the contribu-tion of region II is negligible, ΦV,II≈ 0. In the third region the magnitudes of cavity

radius and velocity are similar to those in region I, but the cavity is collapsing rather than expanding, i.e., r ≈ R0 and ˙r ∼ −U0, leading to ΦV,III∼ R0U0∆zIII.

Inciden-tally, due to the asymmetry between regions I and III their respective contributions are not expected to cancel. The fact that ∆zIII> ∆zI leads to a positive contribution

to the air flow rate. Finally, regions IV and V are the regions where the cavity col-lapses inertially, i.e., the cavity wall accelerates predominantly as a consequence of continuity

r˙r = constant ∼ R0U0 ⇒ r ∼

p

R0U0(tc− t), (2.4)

where tcis the pinch-off time. The time derivative of Eq. (2.4) gives ˙r ∼ −

√

R0U0(tc−

t)−1/2such that r ˙r ∼ −R0U0, independent of z. This now leads to ΦV,IV∼ R0U0∆zIV

and ΦV,V∼ R0U0∆zVrespectively. Combining all of the above we can approximate

Eq. (2.3) as

ΦV = ΦV,I+ ΦV,II+ ΦV,III+ ΦV,IV+ ΦV,V− πR20U0

= R₀U₀(−AI∆zI+ AIII∆zIII+ AIV∆zIV)

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with AI, AIII, AIV, and AVnumerical constants.

The difference between region I to IV and region V lies in the way the vertical length scale scales with the impact speed, i.e., with the Froude number. As demon-strated in [19] and [17], the cavity as a whole, i.e., the pinch-off depth zpinchand the

depth of the disc at the moment of pinch-off zdisc, scale as R0Fr1/2. So the same

scal-ing can also be expected for the vertical length scales ∆zIto ∆zIVintroduced above.

Things are different for region V, close to the pinch off, where there exists a local self-similar coupling between the vertical and the radial cavity dimensions [2, 18, 20]. For this reason ∆zVis expected to be independent of the Froude number, i.e., ∆zV∼ R0.

Inserting the scaling of the vertical length scales into Eq. (2.5) leads to the expected R2₀U0-dependence in all terms and an additional Fr1/2-dependence for the first three

terms only

ΦV = R2₀U0

h

A Fr1/2+ Bi ⇒ Φ_V∗ = A Fr1/2+ B , (2.6) with A and B numerical constants. To test this relation we extended the experiments of Fig. 2.4 by performing boundary integral numerical simulations in order to cover a wide range of Froude numbers†. The obtained results are added to Fig. 2.4 using red diamonds. There is a good agreement with the experimental data, and the non-constant slope is clearly visible. A fit to the simulation data confirms Eq. (2.6) and gives A ≈ 1.23 and B ≈ 1.01.

2.4 Flow visualization

In this Section we perform a direct determination of the air flow velocity by seeding the air with smoke and illuminating with a laser sheet, the results of which we will subsequently compare to the velocities that were determined indirectly and indepen-dently using volume measurements. We will first describe the method and results of the flow visualization that we used to measure the air flow inside the cavity. Be-fore doing the impact experiment we fill the atmosphere above the water surface with small smoke particles. When subsequently the disk is moved down through the water surface, the smoke is dragged along, and fills the cavity created below the surface. We illuminate a thin sheet of the smoke using a 1500 mW diode laser line generator (Magnum II) and record the experiment at a recording rate up to 15 kHz by placing the high speed camera perpendicular to the laser sheet (Fig. 2.6). The smoke consists of small glycerine-based droplets (diameter ∼ 3 µm), produced by a commercially available smoke machine built for light effects in discotheques. A simple analysis shows that the particles are light enough to neglect all inertial effects at least in the

†_{The simulations in Fig. 2.4 are two-phase boundary integral simulations, where close to pinch off}

the compressibility of the gas is taken into account using the one-dimensional compressible Euler equa-tions (briefly discussed in Section 2.5 as type (iii) simulaequa-tions). More details about these simulaequa-tions can be found in [7] and in Chapter 3.

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2.4. FLOW VISUALIZATION 19

laser

disk

camera

linear motor

Figure 2.6: A schematic view of the setup. A laser sheet shines from above on the disk, illuminating the interior of the cavity after the disk has impacted the water surface. We insert smoke in the top part of the container and when the linear motor pulls the disk through the water surface at a constant speed, the smoke is entrained into the cavity.

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Figure 2.7: A snapshot of the cavity with an overlay of a recording of the illuminated smoke. The smoke particles are artificially colored orange in this figure. The size and position of correlation window is indicated by the yellow square.

range of accelerations that we can measure experimentally: At a velocity difference of 10 m/s the Reynolds number is ∼ 2, meaning we can assume Stokes drag. Know-ing the force on the particle as a function of the velocity difference and the mass of the particles, we can calculate the movement of the droplets in an accelerating flow. We find that the particles follow the flow up to 25 m/s with a velocity lag less than 2%.

Correlation technique- We determine the speed of the air in the neck by apply-ing an image correlation velocimetry (ICV) technique [21]. ICV differs from Particle Image Velocimetry (PIV) in the sense that we do not resolve discrete particles in our images, but we correlate smoke patterns instead of smoke particles. Figure 2.7 shows the cavity with the illuminated smoke as an overlay, where the smoke is colored or-ange artificially for clarity. The actual measurements are done on a closer view of the cavity. The correlation is performed on a square correlation window, indicated by the yellow square. The width of the correlation window is 160 pixels, corresponding to 8.8 mm. In the latest stages we switch to a correlation window of 96 pixels (5.3 mm) wide, anticipating on the smaller neck radius. The measurements are insensitive to small changes in the shape, size or position of the correlation window. The size of the window is optimized for quality of the cross correlation.

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struc-2.4. FLOW VISUALIZATION 21 tures visible in the correlation window that move slowly compared to the typical gas velocities that we want to measure. A correlation between two unprocessed images gives a strong correlation peak close to zero because these structures are dominating the image, and thereupon also the cross-correlation. Standard background subtraction is not able to remove these features since, because of their refractive and reflective nature they appear and disappear at unpredictable instances in time and are not sta-tionary. Instead we use the difference between subsequent images, in the following way. We start with three images In, In+1, and In+2. After applying a low pass filter

we create from these three images two new images by subtraction: Jn= In+1− Inand

Jn+1= In+2− In+1. After this we apply a min-max filter [22] to both images, followed

by the cross correlation of Jnand Jn+1. On the result of the correlation we apply a

multiple peak detection to find the highest peak p1 and the second-highest peak p2.

We determine the position of the highest peak with sub-pixel accuracy by a gaussian fitting routine.

A subtraction technique similar to the one that we use here has been used previ-ously for double-frame PIV images [23], where it was found that if the displacement of the particles is too small between a pair of images, the displacement peak in the correlation is biased. This bias is related to the particle size in pixels and the dis-placement in pixels. In our case this length scale does not exist because we do not resolve separate smoke particles in our experimental setup. Instead of calculating the expected bias, we identify biased values by their departure from the global trend of the data (Fig. 2.8, inset). As a remedy for the bias, we artificially increase the displacement by skipping frames. The smaller the velocity, the larger the number of frames we skip. In addition to this we note that the bias is less pronounced compared to the case in [23] because we construct the image pair from three images in stead of two.

The biased data and other spurious data is removed by making an objective se-lection based on the peak-to-peak ratio of the correlation. This ratio is defined as the ratio between the two highest peaks in the correlation: λ = p1/p2. The inset of

Fig-ure 2.8 shows the effectiveness of this selection method. We set λ to values between 3.5 and 5.0, depending on the specific measurement, so that almost all spurious data is removed. Taking higher values for λ removes too much valid data points; lower values allow for too many biased data points.

In Fig. 2.8 we compare the air speed that we measured directly using smoke particles with the air velocity that we calculated indirectly using the change in volume of the cavity, as discussed in the previous Section. The air speeds are plotted versus the neck radius R(t) at pinch-off depth instead of time; time increases from right to left in the Figure, i.e., towards smaller values of R. The blue line is obtained using a polynomial (smoothing) fit to the volume-time data of Fig 2.3, determining the flow rate ΦV(t) from the time derivative of this fit [Eq. (2.2)], and finally dividing

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 −2 0 2 4 6 8 10 12 14 R/R₀ u g (m/s) 0.4 0.6 0.8 0 2 4 6 R/R₀ u g (m/s) biased data

Figure 2.8: The vertical air velocity through the neck as a function of the neck radius R, measured in an experiment with Fr = 5.1 in three different ways: (i) Directly, using smoke particles (diamonds), (ii) Indirectly, using a smoothing polynomial fit to bubble volume of Fig. 2.3 (blue line) , and (iii) Indirectly, using a constant flow rate approximation, determined at pinch off (cf. Fig 2.3, black line). The different colors of the diamonds correspond to different numbers of frames that are skipped in the cross-correlation (see main text). The inset shows the same vertical velocity data measured using method (i) for two different values for the peak-to-peak ratio λ : For λ > 1.5 (orange dots) we find strongly biased data, which are eliminated using a higher threshold (λ > 3.5, black dots).

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2.5. THE ROLE OF COMPRESSIBILITY 23 by πR(t)2to obtain the velocity. We find a very good agreement between the direct (smoke) and the indirect (volume) measurements.

Finally, the black line in Fig. 2.8 is obtained by setting the flow rate to a constant value, namely to that corresponding to the time derivative of the volume curve just before pinch off (the green line in Fig. 2.3). We observe that at early times (large R) there are large deviations from the other two datasets. This stands to reason, since at these times we are still far away from the pinch-off moment, and the gas flow rate in the neck has not yet become (approximately) constant. Close to pinch off however, for R/R0. 0.4, we find that the constant flow rate approximation and the smoothing

fit both provide the same air speed.

2.5 The role of compressibility

The fact that the air flow rate becomes constant together with the surface area of the neck becoming vanishingly small suggests that the velocity in the neck diverges to-wards pinch off. However, as was mentioned in the introduction, a real singularity of the air flow velocity is prevented by compressibility effects. In a previous publication we presented a directly visible effect of the compressed gas flow, namely the upwards motion of the position of the minimum neck radius (Chapter 3). This upwards motion was seen both in experiments and in simulations that take into account the compress-ibility, and is absent in simulations that neglect compressibility. In the same paper we reported that, next to this upward motion of the neck, the extremely fast airflow affects the smoothness of the neck. Especially this last effect is important, since it is in contradiction with the assumptions in theoretical pinch-off models where the neck is assumed to be slender [18, 20].

The question that we intend to answer in the present Section is how the effects of compressibility show up in the measurement of the cavity volume and the air flow rate that can be deduced from it, as was presented in Section 2.3 of this work. More specifically we will investigate the position of the stagnation point of the flow in the cavity (see below) and the air flow rate towards the pinch-off moment. Following the method we used in Chapter 3, we will compare our experimental results with three different types of boundary integral simulations: (i) a single phase version, in which only the water phase is resolved, (ii) a two-phase version where both the liquid and the gas flow are resolved as incompressible inviscid media, and (iii) a compressible gas version where the compressibility of the gas phase is taken into account by substituting the incompressible axisymmetric gas phase equations by one-dimensional compressible Euler equations at that moment during the collapse when compressibility effects start to become significant. More details about the numerical method can be found in [7] and in Chapter 3.

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approxi-mately the same speed as the disc, whereas simultaneously, towards closure, the air in the neck is moving upwards. This implies that somewhere in between there will be a stagnation point. We will estimate the location of this stagnation point as follows: The first step is to extend the analysis of Section 2.3, where we tracked the volume below the pinch-off depth in time, to any depth z below the pinch-off point. For every depth z this will provide us with a curve similar to that in Fig. 2.3 and by determining the time coordinate of the maximum we find the time tstag at which the averaged ‡

flow rate (∼ ˙V) at that depth z ≡ zstagis zero. This point we then interpret as the

loca-tion of the stagnaloca-tion point zstag(tstag), which involves the assumption that close to the

pinch-off moment the flow in the neck region becomes predominantly homogeneous and vertical. In Fig. 2.9 we plot the measured location of zstag for three different

realizations of an experiment with a radius of 2 cm and an impact speed of 1 m/s. A difference with the actual location of the stagnation point is therefore expected for high gas velocities (i.e., small neck radii). When we compare the experiments to a two-phase incompressible boundary integral simulation [type (ii)] (red line in Fig. 2.9, we find a considerable discrepancy between the two for small values of the neck radius R. If we however use the compressible version of the simulation [type (iii)], the agreement becomes much better (red line in Fig. 2.9, confirming the impor-tance of compressibility in this limit. Note that experiments and both simulations do converge for larger values of R, where compressibility effects play no role.

The agreement is not perfect however, which can partly be traced back to the technical difficulty of obtaining reliable values for zstag from the experiment (which

reflects in the large spread between the three different realizations) and partly to the fact that its determination neglects compressibility in a subtle way: Although in the experimental data compressibility is of course necessarily reflected in the shape of the cavity, the method of obtaining the air flow rate from it (namely by determining the time rate of change of the cavity volume) neglects compressibility in the air phase.

Air flow rate- As explained in the introduction of this Section, we can compare the air flow rate in the neck in experiment and simulation directly, by comparing the experimental velocities (cf. Section 2.4) as was done in Chapter 3, but also indirectly, by using the volume analysis of Section 2.3 both in experiment and simulation. This second method, the results of which will be presented now, enables us to distinguish the effect the compressibility of the air has on the cavity wall (which is included in the analysis) from the pure compressibility of the flow (which is not included).

To do so it is convenient to from now on distinguish the true air flow rate from the derived air flow rate, i.e., the one obtained from the time rate of change of the cavity volume. In Fig. 2.10a we plot the non-dimensionalized experimental derived air flow

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2.5. THE ROLE OF COMPRESSIBILITY 25 0 0.005 0.01 0.015 0.02 −0.03 −0.02 −0.01 0 0.01 R (m) z c − z stag (m)

Figure 2.9: The location of the stagnation point zstagwith respect to that of the

pinch-off point zcas a function of the neck radius R. Note that when that the stagnation point

lies below the pinch-off point zc− zstagis negative. Time increases from right to left

(decreasing R). The dots are experimental data, obtained by volume measurements of four different experiments, where each color corresponds to a different experiment. All experiments were performed with disk radius R0= 2.0 cm and impact speed U0=

1.0 m/s, i.e., Fr = 5.1. The green line is the result of a two-phase boundary integral simulation without taking compressibility into account [type (ii)]. The red line is obtained by a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)].

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0 0.5 1 −5 0 5

R/R

_o

Φ

V * (a) Two−phase compressible Two−phase incompressible Single phase Experimental data 0 0.5 1 −5 0 5

R/R

_o (b) Φ* Φ V *

Figure 2.10: (a) The dimensionless derived air flow rate Φ_V∗ = ΦV/(R2₀U0) (from the

time rate of change of the cavity volume) as a function of the dimensionless neck radius R/R0 in an impact experiment with disc radius R0= 2 cm and impact speed

U0= 1 m/s (Fr = 5.1). The black dots represent experimental data. The red line is

obtained using a one-phase simulation [type(i)], which excludes the air phase. The green line is a two-phase boundary integral simulation without compressibility [type (ii)]. Finally, the blue line is the result of a two-phase boundary integral simulation which includes a compressible gas phase [type (iii)]. (b) Comparison of the dimen-sionless derived air flow rate Φ_V∗ [blue line; the same curve as in (a)] and the true air flow rate Φ∗, both plotted versus R/R0. The two curves diverge from each other

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2.5. THE ROLE OF COMPRESSIBILITY 27 rate in the neck,

ΦV∗ ≡ ΦV R2₀U0 ≡ − 1 R2₀U0 dV dt , (2.7)

as a function of the dimensionless neck radius R(t)/R0(black dots), again for Fr =

5.1. Repeating the experiment results in an uncertainty in the magnitude of ΦV∗

(cor-responding to the spread of the experimental data in Fig. 2.4), but the behavior as a function of time is always the same: The derived air flow rate in the neck reaches a maximum, and approaches a finite value towards the moment of pinch-off. We compare this result with those of the three different types of boundary integral simu-lations:

The one-phase code [type (i)] predicts a steadily increasing derived air flow rate, which seems to level off to a constant value towards pinch off (R/R0→ 0). This is

the red line in Fig. 2.10(a).

The two-phase incompressible version [type (ii)] predicts a maximum at a loca-tion which is reasonably comparable to the experimental one, but after that decreases toward zero at the pinch-off moment (the green curve in Fig. 2.10(a)). Since both phases are incompressible, this stands to reason: The pressure in the cavity rises in-stantly because of the divergence of the air velocity ug in the shrinking neck. This

pressure decelerates the cavity wall, which in turn decreases the derived air flow rate, which should go to zero in the R/R0→ 0-limit: In the context of incompressible flow,

a finite derived air flow rate would result in an infinite air velocity in the neck and consequently an infinite pressure within the cavity. Here it is good to note that for this two-phase incompressible code the derived and true air flow rates are actually identical, due to the incompressibility of the air phase.

The two-phase compressible simulation [type (iii)] also predicts a maximum for Φ∗_V, at a location similar to the two-phase incompressible code and the experiment, but then decreases to a finite value for R/R0→ 0, just like the experiment. Clearly,

and in contrast with the other two versions of the simulation which behave poorly, the agreement with the experiments is qualitatively very good and quantitatively sat-isfactory. All three types of simulations and the experiments all converge for larger R/R0≈O(1), which is expected since airflow effects (let alone compressibility of

the air phase) are small or even negligible in that regime.

The final question that we want to address is the difference between the true air flow rate (which incorporates all compressibility effects) and the derived one (which only includes the effects of compressibility on the cavity wall). In experiment it is impossible to obtain the first quantity at the required precision, because its determi-nation includes measurement errors in both air velocity ug and neck radius R. The

two-phase compressible simulation technique however does offer a way to look at this difference: In Fig. 2.10(b) we compare the derived air flow rate ΦV∗ (the same

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curve as the blue one in Fig. 2.10(a)) to the true air flow rate Φ∗(green curve), which is calculated from ug(t) and R(t) as

Φ∗≡ Φ R2 0U0 = 1 R2 0U0 ug(t) π R(t)2, (2.8)

both as a function of the dimensionless cavity radius R(t)/R0. Clearly the two curves

coincide above R/R0≈ 0.2, but start to depart from one another below this value,

indicating that here the compressibility of the air itself becomes significant, in good agreement with what we concluded from the previous plot (Fig. 2.10(a)). We observe that the true air flow rate goes to zero for R/R0→ 0 (and incidentally not quite unlike

the two-phase incompressible curve (green) in Fig. 2.10(a)). This is of course what should happen, since the gas velocity in the neck needs to remain finite at all times. The difference between the two curves is the rate at which the gas in the cavity is compressed.

2.6 Conclusions

We have measured the air flow inside the neck of a collapsing cavity that was created by the impact of a circular disc on a water surface. More specifically we have per-formed and compared two types of experiments: First we did indirect measurements, using the time rate of change of the cavity volume as a measure for the air flow rate in the neck, thereby neglecting compressibility of the air inside the cavity. Secondly we performed direct measurements of the velocity in the neck of the cavity using image correlation velocimetry. Numerical boundary integral simulations of three different types have been used to evaluate and discuss our experimental findings.

For the complete experimentally available range of Froude numbers we showed that there is a very good agreement between the indirectly measured air flow rate and the boundary integral simulations. With the simulations we were able to extend the range of experimentally attainable Froude numbers, which revealed that the air flow rate is not a pure power-law of the Froude number. We formulated an analytical argument revealing that the dimensionless air flow rate should scale as AFr1/2+ B. Such a scaling compares well with experiments and simulations for A ≈ 1.23 and B≈ 1.01.

By performing careful image correlation velocimetry experiments with a smoke-filled cavity we have been able to directly measure the air flow for relatively low air speeds, corresponding to R/R0≥ 0.3. In this region we found excellent agreement

with the gas velocities that we calculated from the indirect measurements of the air flow rate and the neck radius R(t).

Due to the very high air speed close to the moment of pinch-off (R/R0≤ 0.2)

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2.6. CONCLUSIONS 29 by comparing experimental results to three types of numerical simulations: (i) one-phase boundary integral simulations, (ii) two-one-phase boundary integral simulations with an incompressible gas-phase, and (iii) a compressible gas version of the second type of simulations that include the gas phase as a compressible fluid. We analyze the time evolution of both the location of the stagnation point in the gas flow and the derived air flow rate and explain our experimental observations in terms of the three types of simulations. The main conclusion is that the behavior that we observe in the experiments can only be reproduced by the simulations if compressibility is taken into account.

Appendix: Derivation of the scaling law for Φ

_V∗

In this Appendix we show that the main result of § 2.3.2 can also be derived in a slightly more rigorous manner, starting from the description of the cavity proposed in [17]. The starting point is the two-dimensional Rayleigh equation for the cav-ity wall r(z,t), which originates from integrating the Euler equations in uncoupled horizontal layers of flow from some far away point R∞to the cavity wall

log (r/R∞) d dt(r ˙r) + 1 2˙r 2_{= gz} _(2.9)

in which ˙r = ∂ r/∂ t and g is the acceleration of gravity.

This equation is solved in two different limits to describe the different regions in Fig. 2.11. The first one is to describe region A and B, taking for every depth z the moment tM(z) of maximum expansion as a reference point. With r(tM) = RM(z),

˙r(tM) = 0, we can neglect the second term in Eq. (2.9) and replace the slowly varying

logarithm in the first term by a constant§β ≡ log(RM/R∞)¶and solve

r(z,t)2= RM(z)2−

gz

β (t − tM(z))

2_. _(2.10)

In [17] it was shown that

tM(z) =

z

U₀+ αexpaβexpa R0U0

gz , (2.11)

in which the first term represents the time span needed to arrive at depth z and the second the amount of time to expand to the maximum radius. Here αexpaβexpa is a

constantk.

§_{Although strictly speaking R}

M(z) is a function of z, it is slowly varying and can approximated by a

constant when the logarithm of this quantity is taken.

¶_{The constant β is different in the expansion β}

expaand the contraction phase (βctra. k_{The nomenclature of the constants is chosen such as to be consistent with [17].}

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Figure 2.11: For the more rigorous derivation in this Appendix, the cavity close to the pinch-off moment which was divided in five different regions in Fig. 2.5 needs be redivided into four regions: The expansion region (A), between the location of the disc zdiscand the location of the maximum zM, where the cavity expands against

hydrostatic pressure; the contraction region (B), between zMand the point zcrosswhere

the cavity reaches the disc radius again where the hydrostatic pressure approximation [Eq. (2.10)] is matched to the inertial approximation [Eq. (2.12)]; the collapse region (C) between zcrossand z∗, characterized by continuity; and the self-similarity region

(D), between z∗and the pinch off location zc, which is in addition characterized by a

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2.6. CONCLUSIONS 31 The second approximate solution corresponds to the small R limit in the collapse regions C and D of Fig. 2.11, in which both the driving pressure gz and the inertial term ˙r2/2 can be considered small when | log(r/R∞)| 1/2. This then leads to

d

dt(r ˙r) = 0 which is readily solved to give:

r(z,t)2= 2αctraR0U0(tcoll(z) − t) , (2.12)

in which αctra is a constant and tcoll(z) is the (virtual) closure time of the cavity at

depth z. At any depth z the approximate solutions are tied together at the maximum (where a solution Eq. (2.10) with β = βexpais matched to a solution with β = βctra)

and at the moment tcross(z) when r(z,t) = R0again. Here, the solution Eq. (2.10) with

β = βctrais matched to Eq. (2.12). More details can be found in [17].

The quantity we want to calculate is Eq. (2.3), which contains the time derivatives of Eqs. (2.10) and (2.12), which are:

d dt r(z,t) 2 = −2gz β (t − tM(z)) (regionA, B) d dt r(z,t) 2 = −2αctraR0U0 (regionC, D) . (2.13)

which subsequently need to be evaluated at the moment of pinch-off t = tc, for which

it was derived in [17] that it is independent of the impact speed: tc= C2

p

R0/g ∗∗.

Inserting this expression together with Eq. (2.11) into the first Eq. (2.13) gives d dt r(z,tc) 2₌ _(2.14) −2gz β C2 s R0 g − z U0 − αexpaβexpa R0U0 gz ! .

Finally we need to integrate the second Eq. (2.13) and Eq. (2.14) over z between zdisc

and zc. This is a straightforward calculation which gives the following lengthy result Z z_disc zc d dt r(z,tc) 2_{dz =} _(2.15) − C2 βexpa p R0g z2disc− z2M + 2 3βexpa z3_disc− z3 M + 2αexpaR0U0(zdisc− zM) − C2 βctra p R0g z2M− z2cross + 2 3βctra z3_M− z3

cross + 2αexpaR0U0(zM− zcross) −

2αctra(zcross− z∗) − 2αctra(z∗− zc) .

∗∗_{The constant C}

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We now use that all length scales zdisc, zM, zcross, and z∗scale as R0Fr1/2, except for

the difference (z∗− zc), which due to the self-similarity in the neck radius scales as

R0. This means that the above Eq. (2.15) has the folowing form Z z_disc zc d dt r(z,tc) 2_{dz =} _(2.16) −κ1 p R0g R20Fr+ κ2g R30Fr3/2+ κ3R20U0Fr1/2− κ4R20U0,

in which κ1-κ4are positive numerical constants, which depend on the α’s, β ’s and

the proportionality constants in the scaling laws for the length scales zdisc, zM, zcross,

z∗, and (z∗− zc) . By writing

√

R₀g= U0Fr−1/2 in the first two terms we finally

obtain: Z zdisc zc d dt r(z,tc) 2_{dz =} _(2.17) (−κ1+ κ2+ κ3) R20U0Fr1/2 − κ4R20U0.

If we now insert the above result in Eq. (2.3) we obtain

ΦV = −

Z z_disc

zc

2π r(z,t) ˙r(z,t, )dz − πR2₀U0 (2.18)

= π (κ1− κ2− κ3) R20U0Fr1/2+ π(κ4− 1)R20U0

which then leads to

Φ∗V ≡

ΦV

R2₀U0

= A Fr1/2+ B , (2.19)

with A ≡ π(κ1− κ2− κ3) and B ≡ π(κ4− 1). The shape of this equation is identical

to Eq. (2.6) we derived in a more heuristic manner in Subsection 2.3.2.

References

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Free surface flow focusing

Free surface flow focusing

FREE SURFACE FLOW FOCUSING

Contents

1

Introduction

References

2

Air flow in a collapsing cavity

∗

2.1

Introduction

2.2

Experimental setup

5 cm

2.3

Geometric approach

2.4

Flow visualization

laser

disk

camera

linear motor

2.5

The role of compressibility

R/R

Φ

R/R

2.6

Conclusions

Appendix: Derivation of the scaling law for Φ

References